Graduate Texts in Mathematics 140 Editorial Board S Axler F W Gehring K A Ribet Springer-Verlag Berlin Heidelberg GmbH Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 TAKEUTIIZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nd ed HIL TON/ST AMMBACH A Course in Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician 2nd ed HUGHESIPIPER Projective Planes J.-P SERRE A Course in Arithmetic TAKEUTIIZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable I 2nd ed BEALS Advanced Mathematical Analysis ANDERSONlFuLLER Rings and Categories of Modules 2nd ed GOLUBITSKY/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMos A Hilbert Space Problem Book 2nded HUSEMOLLER Fibre Bundles 3rd cd HUMPHREYS Linear Algebraic Groups BARNESIMACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKI/SAMUEL Commutative Algebra Vol I ZARISKIISAMUEL Commutative Algebra Vol II JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nded 35 ALEXANDERIWERMER Several Complex Variables and Banach Algebras 3rd ed 36 KELLEy/NAMIoKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERTIFRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed 41 APOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 J.-P SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LoEvE Probability Theory I 4th ed 46 LOEVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHSlWu General Relativity for Mathematicians 49 GRUENBERGIWEIR Linear Geometry 2nded 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRA VERIWATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BRowNIPEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELLlFox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy Theory 62 KARGAPOLOVIMERLZJAKOV Fundamentals of the Theory of Groups 63 BOLLOBAS Graph Theory (continued after index) Jean-Pierre Aubin Optima and Equilibria An Introduction to Nonlinear Analysis Translated from the French by Stephen Wilson With 28 Figures Second Edition 1998 , Springer Jean-Pierre Aubin Reseau de Recherche Viabilite, Jeux, Controle 14, rue Domat 75005 Paris, France Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132, USA axler@sfsu.edu F W Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109, USA fgehring@math.1sa.umich.edu K A Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840, USA ribet@math.berkeley.edu Titles of the French original editions: L'analyse non lineaire et ses motivations economiques © Masson Paris 1984, Exercices d' analyse non lineaire © Masson Paris 1987 Mathematics Subject Classification (2000): 91A, 9IB, 65K, 47H, 47NlO, 49J, 49N Corrected 2nd printing 2003 ISSN 0072-5285 ISBN 978-3-642-08446-1 ISBN 978-3-662-03539-9 (eBook) DOI 10.1007/978-3-662-03539-9 Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Anbin, Jean-Pierre: Optima and equilibria: an introduction to nonlinear analysis / Jean-Pierre Aubin Trans! from the French by Stephen Wilson - ed - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 1998 (Graduate texts in mathematics; 140) This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable for prosecution under the German Copyright Law http://www.springer.de © Springer-Verlag Berlin Heidelberg 1993, 1998 Originally published by Springer-Verlag Berlin Heidelberg New York in 1993, 1998 Softcover reprint of the hardcover 2nd edition 1998 The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: Camera-ready by the translator using Springer TEX macropackage Cover design: design & production GmbH, Heidelberg SPIN 10988749 4113111 - 5432 - Printed on acid-free paper This book is dedicated to Alain Bensoussan, Ivar Ekeland, Pierre-Marie Larnac and Francine Roure, in memory of the adventure which brought us together more than twenty years ago to found the U.E.R and the Centre de Recherche de Mathematiques de la Decision (CEREMADE) Jean-Pierre Aubin Doubtless you have often been asked about the purpose of mathematics and whether the delicate constructions which we conceive as entities are not artificial and generated at whim Amongst those who ask this question, I would single out the practical minded who only look to us for the means to make money Such people not deserve a reply Henri Poincare La Valeur de La Science Chapter V In his use of mathematical techniques to study general economic phenomena relating to countries or individuals Mr Leon Walras has truly instituted a science Charles Peguy Un economiste socialiste, Mr Leon Walras La Revue Socialiste, no 146, 1897 It may be that the coldness and the objectivity for which we often reproach scientists are more suitable than feverishness and subjectivity as far as certain human problems are concerned It is passions which use science to support their cause Science does not lead to racism and hatred Hatred calls on science to justify its racism Some scientists may be reproached for the ardour with which they sometimes defend their ideas But genocide has never been perpetrated in order to ensure the success of a scientific theory At the end of this the XXth century, it should be clear to everyone that no system can explain the world in all its aspects and detail Quashing the idea of an intangible and eternal truth is possibly not the least claim to fame of the scientific approach Fran~ois Jacob Le Jeu des possibL(~s Fayard (1981) p 12 I enjoy talking to great minds and this is a taste which I like to instil in my students I find that students need someone to admire; since they cannot normally admire their teachers because their teachers are examiners or are not admirable, they must admire great minds while, for their part, teachers must interpret great minds for their students Raymond Aron Le Spectateur engage Julliard (1981) p 302 Foreword By Way of Warning As in ordinary language, metaphors may be used in mathematics to explain a given phenomenon by associating it with another which is (or is considered to be) more familiar It is this sense of familiarity, whether individual or collective, innate or acquired by education, which enables one to convince oneself that one has understood the phenomenon in question Contrary to popular opinion, mathematics is not simply a richer or more precise language Mathematical reasoning is a separate faculty possessed by all human brains, just like the ability to compose or listen to music, to paint or look at paintings, to believe in and follow cultural or moral codes, etc But it is impossible (and dangerous) to compare these various faculties within a hierarchical framework; in particular, one cannot speak of the superiority of the language of mathematics Naturally, the construction of mathematical metaphors requires the autonomous development of the discipline to provide theories which may be substituted for or associated with the phenomena to be explained This is the domain of pure mathematics The construction of the mathematical corpus obeys its own logic, like that of literature, music or art In all these domains, a temporary aesthetic satisfaction is at once the objective of the creative activity and a signal which enables one to recognise successful works (Likewise, in all these domains, fashionable phenomena - reflecting social consensus - are used to develop aesthetic criteria) That is not all A mathematical metaphor associates a mathematical theory with another object There are two ways of viewing this association The first and best-known way is to search for a theory in the mathematical corpus which corresponds as precisely as possible with a given phenomenon This is the domain of applied mathematics, as it is usually understood But the association is not always made in this way; the mathematician should not be simply a purveyor of formulae for the user Other disciplines, notably physics, have guided mathematicians in their selection of problems from amongst the many arising and have prevented them from continually turning around in the same circle by presenting them with new challenges and encouraging them to be daring and question the ideas of their predecessors These other disciplines may also pro- VIII Foreword vide mathematicians with metaphors, in that they may suggest concepts and arguments, hint at solutions and embody new modes of intuition This is the domain of what one might call motivated mathematics from which the examples you will read about in this book are drawn You should soon realize that the work of a motivated mathematician is daring, above all where problems from the soft sciences, such as social sciences and, to a lesser degree, biology, are concerned Many hours of thought may very well only lead to the mathematically obvious or to problems which cannot be solved in the short term, while the same effort expended on a structured problem of pure or applied mathematics would normally lead to visible results Motivated mathematicians must possess a sound knowledge of another discipline and have an adequate arsenal of mathematical techniques at their fingertips together with the capacity to create new techniques (often similar to those they already know) In a constant, difficult and frustrating dialogue they must investigate whether the problem in question can be solved using the techniques which they have at hand or, if this is not the case, they must negotiate a deformation of the problem (a possible restructuring which often seemingly leads to the original model being forgotten) to produce an ad hoc theory which they sense will be useful later They must convince their colleagues in the other disciplines that they need a very long period for learning and appreciation in order to grasp the language of a given theory, its foundations and main results and that the proof and application of the simplest, the most naive and the most attractive results may require theorems which may be given in a number of papers over several decades; in fact, one's comprehension of a mathematical theory is never complete In a century when no more cathedrals are being built, but impressive skyscrapers rise up so rapidly, the profession of the motivated mathematician is becoming rare This explains why users are very often not aware of how mathematics could be used to improve aspects of the questions with which they are concerned When users are aware of this, the intersection of their central areas of interest with the preoccupations of mathematicians is often small - users are interested in immediate impacts on their problems and not in the mathematical techniques that could be used and their relationship with the overall mathematical structure It is these constraints which distinguish mathematicians from researchers in other disciplines who use mathematics, with a different time constant It is clear that the slowness and the esoteric aspect of the work of mathematicians may lead to impatience amongst those who expect them to come up with rapid responses to their problems Thus, it is vain to hope to pilot the mathematics downstream as those who believe that scientific development may be programmed (or worse still, planned) may suggest In Part I, we shall only cover aspects of pure mathematics (optimisation and nonlinear analysis) and aspects of mathematics motivated by economic theory and game theory It is still too early to talk about applying mathematics to economics Several fruitful attempts have been made here and there, but mathematicians are a long way from developing the mathematical techniques Foreword IX (the domains of pure mathematics) which are best adapted to the potential applications However, there has been much progress in the last century since pioneers such as Quesnais, Boda, Condorcet, Cournot, Auguste and Leon Walras, despite great opposition, dared to use the tools of mathematics in the economic domain Brouwer, von Neumann, Kakutani, Nash, Arrow, Debreu, Scarf, Shapley, Ky Fan and many others all contributed to the knowledge you are about to share You will surely be disappointed by the fact that these difficult theorems have little relevance to the major problems facing mankind But, please don't be impatient, like others, in your desire for an overall, all-embracing explanation Professional mathematicians must be very humble and modest It is this modesty which distinguishes mathematicians and scientists in general from prophets, ideologists and modern system analysts The range of scientific explanations is reduced, hypotheses must be contrasted with logic (this is the case in mathematics) or with experience (thus, these explanations must be falsifiable or refutable) Ideologies are free from these two requirements and thus all the more seductive But what is the underlying motivation, other than to contribute to an explanation of reality? We are brains which perceive the outside world and which intercommunicate in various ways, using natural language, mathematics, bodily expressions, pictorial and musical techniques, etc It is the consensus on the consistency of individual perceptions of the environment, which in some way measures the degree of reality in a given social group Since our brains were built on the same model, and since the ability to believe in explanations appears to be innate and universal, there is a very good chance that a social group may have a sufficiently broad consensus that its members share a common concept of reality But prophets and sages often challenge this consensus, while high priests and guardians of the ideology tend to dogmatise it and impose it on the members of the social group (Moreover, quite often prophets and sages themselves become the high priests and guardians of the ideology, the other way round being exceptional.) This continual struggle forms the framework for the history of science Thus, research must contribute to the evolution of this consensus, teaching must disseminate it, without dogmatism, placing knowledge in its relative setting and making you take part in man's struggle, since the day when Homo sapiens, sapiens But we not know what happened, we not know when, why or how our ancestors sought to agree on their perceptions of the world to create myths and theories, when why or how they transformed their faculty for exploration into an insatiable curiosity, when, why or how mathematical faculties appeared, etc It is not only the utilitarian nature (in the short term) which has motivated mathematicians and other scientists in their quest We all know that without this permanent, free curiosity there would be no technical or technological progress X Foreword Perhaps you will not use the techniques you will soon master and the results you will learn in your professional life But the hours of thought which you will have devoted to understanding these theories will (subtly and without you being aware) shape your own way of viewing the world, which seems to be the hard kernel around which knowledge organizes itself as it is acquired At the end of the day, it is at this level that you must judge the relevance of these lessons and seek the reward for your efforts 17.9 Set-valued Maps and the Existence of Zeros and Fixed Points 421 We consider two matrices F and G from IRn to IRm satisfying (i) the coefficients % of G are non-negative; (ii) Vi (iii) Vj = 1, ,m, = 1, L7=1 gij > 0; ,n, L7=1 fij > O Then there exist x E Mn, p E Mm and > such that oFx S Gx of' p- > G'-p o(p, Fx) = (p, Gx) (i) (ii) (iii) Moreover, for all fL > a and all y E lnt IR~, there exists x E IR~ such that ILFx - Gx S y (18) Suppose that F is a mapping from Mn to IRn satisfying (i) the components fi of F are convex and lower semi-continuous; (ii) ::Jp E Mn n lnt(lR~) such that Vx E Mn, (p, F(x)) > 0; (iii) if Xi = then fJr) S O Suppose that G is another mapping from Mn to IRn satisfying (i) the components (ii) Vx E lVr, Vi = gi of G are concave and lower semi-continuous; 1, ,n, gi(X) > O We consider the number > defined by (*) (above) Then there exist x E Mn n lnt(IR~) and p E Mn n lnt(IR~) such that (i) (ii) If fL > and y E such that Vx E of(x) = G(x) Mn, (p, G(x) - of(x)) SO lnt(IR~) are given, then there exist (3 > and x E Mn fLF(x) - G(x) = (3y (19) 422 17 Compendium of Results Perron-Frobenius Theorem Suppose that G is a positive matrix (a) (b) (c) (d) G has a strictly positive eigenvalue J and an associated eigenvector i; with strictly positive components J is the only eigenvalue associated with an eigenvector of Mn J is greater than or equal to the absolute value of all other eigenvalues of G The matrix JL - G is invertible and (JL - Gr is positive if and (9.20) only if JL > J We consider a mapping H from IR~ to IRn satisfying (i) the components hi of H are convex, positively 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